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### Classification of Linear Codes by Extending Their Residuals [chapter]

Stefka Bouyuklieva, Iliya Bouyukliev
2020 Lecture Notes in Computer Science
An approach for classification of linear codes with given parameters starting from their proper residual codes or subcodes is presented.  ...  As an application, the nonexistence of binary [41, 14, 14] codes is proved.  ...  We have to prove that (1) any [n, k, d] q code with the needed dual distance is equivalent to a code in the set M , and (2) the codes in M are not equivalent. (1) Let C be an [n, k, d] q code with dual  ...

### Classification of the extremal formally self-dual even codes of length 30

Stefka Bouyuklieva, Iliya Bouyukliev
2010 Advances in Mathematics of Communications
Let C be a binary [n, k, d] code and let c be a codeword of weight d. Then As the residual code can be considered as a punctured code on a set of coordinates, its dual distance is at least d ⊥ .  ...  Theorem 2.  Let C be a binary [n, k, d] code and let c be a codeword of weight Actually, we consider a residual code with respect to a codeword of minimum weight.  ...  If C is a linear [n, k, d] code with dual distance d ⊥ and T is a set of t coordinates, then the dual distance of its punctured code C T is at least d ⊥ .  ...

### Results on Binary Linear Codes With Minimum Distance 8 and 10

Iliya Georgiev Bouyukliev, Erik Jacobsson
2011 IEEE Transactions on Information Theory
All codes with minimum distance 8 and codimension up to 14 and all codes with minimum distance 10 and codimension up to 18 are classified.  ...  Primarily two algorithms considering the dual codes are used, namely extension of dual codes with a proper coordinate, and a fast algorithm for finding a maximum clique in a graph, which is modified to  ...  Then Res(C, c) is an [n−w, k−1, d ′ ] code with d ′ ≥ d−w+⌈w/2⌉, and on the dual distance by Proposition 4 Suppose C is a binary [n, k, d] code with dual distance d ⊥ , c ∈ C, and the dimension of Res(  ...

### Optimal subcodes of formally self-dual codes and their optimum distance profiles

Finley Freibert, Jon-Lark Kim
2013 Applicable Algebra in Engineering, Communication and Computing
In this paper we show that some optimal formally self-dual codes actually contain optimal subcodes by computing the optimum distance profiles (ODPs) of linear codes.  ...  A (binary) formally self-dual code is a linear code whose weight enumerator is equal to that of its dual. Little is known about the existence of optimal subcodes of formally self-dual codes.  ...  minimum distances than the minimum distances of self-dual codes.  ...

### On the classification of quaternary optimal Hermitian LCD codes [article]

2021 arXiv   pre-print
We propose a method for a classification of quaternary Hermitian LCD codes having large minimum weights.  ...  As an example, we give a classification of quaternary optimal Hermitian LCD codes of dimension 3.  ...  [n, k, d] codes with dual distances d ⊥ ≥ 2.  ...

### Characterization and classification of optimal LCD codes [article]

Makoto Araya, Masaaki Harada, Ken Saito
2021 arXiv   pre-print
Linear complementary dual (LCD) codes are linear codes that intersect with their dual trivially. We give a characterization of LCD codes over 𝔽_q having large minimum weights for q ∈{2,3}.  ...  Moreover, we give a complete classification of optimal LCD [n,k] codes over 𝔽_q for (q,k) ∈{(2,3), (2,4), (3,2),(3,3)}.  ...  Theorem 4.5 (i) claims that there is a one-to-one correspondence between equivalence classes of LCD [n, k, d] codes over F q with dual distances d ⊥ ≥ 2 and equivalence classes of LCD [q · r q,n,k,d ,  ...

### A simple combinatorial treatment of constructions and threshold gaps of ramp schemes

Maura B. Paterson, Douglas R. Stinson
2013 Cryptography and Communications
Finally, as an immediate consequence of these results, we provide a new explicit bound on the the minimum length of a code having a specified distance and dual distance.  ...  We then generalise a construction method for ramp schemes employing error-correcting codes so that it can be applied using nonlinear (as well as linear) codes.  ...  [8, 9] Suppose C is a code of length n, on an alphabet of size q, having dual distance d * .  ...

### On components of a Kerdock code and the dual of the BCH code C_1,3 [article]

I. Yu. Mogilnykh, F. I. Solov'eva
2018 arXiv   pre-print
In the paper we investigate the structure of i-components of two classes of codes: Kerdock codes and the duals of the primitive cyclic BCH code with designed distance 5 of length n=2^m-1, for odd m.  ...  We prove that for any admissible length a punctured Kerdock code consists of two i-components and the dual of BCH code is a i-component for any i.  ...  From (5) we see that any codeword of K ′′ n/2 is at distance d − 2 from at least one codeword of K d−2 and a codeword of K ′′ n−d−2 is at distance d − 2 from at least one codeword of K ′′ n/2 .  ...

### An improved upper bound on self-dual codes over finite fields GF(11), GF(19), and GF(23) [article]

Whan-Hyuk Choi, Jon-Lark Kim
2021 arXiv   pre-print
Consequently, we improve bounds on the highest minimum distance of self-dual codes, which have not been significantly updated for almost two decades.  ...  They all have new parameters with respect to self-dual codes.  ...  On the other hand, self-dual codes have been the subject of much interest and are regarded as one of the most important classes of error-correcting codes.  ...

### Weight distribution of cosets of small codes with good dual properties [article]

Louay Bazzi
2017 arXiv   pre-print
The bilateral minimum distance of a binary linear code is the maximum d such that all nonzero codewords have weights between d and n-d.  ...  Let Q⊂{0,1}^n be a binary linear code whose dual has bilateral minimum distance at least d, where d is odd.  ...  previous works is the requirement on the dual code to have good distance properties.  ...

### Bounds for Self-Dual Codes Over Z4

Eric Rains
2000 Finite Fields and Their Applications
The proof of these bounds relies on a reduction to a problem of binary codes, namely that of bounding the minimum dual distance of a doubly even binary code. Academic Press  ...  New bounds are given for the minimal Hamming and Lee weights of self-dual codes over 9 .  ...  For instance, for n"24m#1, we have 6m#1 coe\$cients and U (d/4 V#U d!1)/4 V#(d!1) constraints. For d"4m, we have only 6m!  ...

### On the covering radius of small codes versus dual distance [article]

Louay Bazzi
2018 arXiv   pre-print
Tietäväinen's upper and lower bounds assert that for block-length-n linear codes with dual distance d, the covering radius R is at most n/2-(1/2-o(1))√(dn) and typically at least n/2-Θ(√(dnn/d)).  ...  Namely, if the dual distance d = o(n), then for sufficiently large d, almost all points can be covered with radius R≤n/2-Θ(√(dnn/d)).  ...  Namely, if µ is uniformly distributed on an F 2 -linear code C ⊂ F n 2 , then µ being k-wise independent is equivalent to C having dual minimum distance at least k + 1.  ...

### Optimal Binary LCD Codes [article]

Stefka Bouyuklieva
2021 arXiv   pre-print
Linear complementary dual codes (or codes with complementary duals) are codes whose intersections with their dual codes are trivial. These codes were first introduced by Massey in 1964.  ...  We also present some inequalities for the largest minimum weight d_LCD(n,k) of binary LCD [n,k] codes for given length n and dimension k.  ...  Acknowledgements I would like to thank Petar Boyvalenkov for focusing my attention to LCD codes and introducing me to the main literature related to this type of codes.  ...

### Some Results for Linear Binary Codes With Minimum Distance\$5\$and\$6\$

I. Bouyukliev, Z. Varbanov
2005 IEEE Transactions on Information Theory
We prove that a linear binary code with parameters [34, 24, 5] does not exist. Also, we characterize some codes with minimum distance 5.  ...  Let C r = [n − d, k − 1, ≥ d/2 ] be its residual Res d (C) code and let a code with parameters [n−d+1, k −1, d ] and dual distance d ⊥ does not exist.  ...  It is known that C ⊥ is an [n, n − k, d ⊥ ] code. Also, d ⊥ is called dual distance of the code. Definition 2 Let G be a generator matrix of a linear binary [n, k, d] code C.  ...

### Neighborhoods of binary self-dual codes [article]

Carolin Hannusch, S. Roland Major
2022 arXiv   pre-print
Further, we show that for codelength divisible by 8 such a neighborhood consists of three self-dual codes, two of them are doubly-even and one is always singly-even.  ...  In this paper, we introduce the neighborhood of binary self-dual codes.  ...  If a code C is a k-dimensional subspace of F n 2 with minimum distance d, then we say that C is an (n, k, d)-code. Self-dual binary codes can be classified into Type I and Type II codes  .  ...
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